[FOM] weak extension of ZFC

Has any thought been given towards `philosophical' justifications
for adding to ZFC the axiom E:
the exponential function is increasing
(for all $\lambda$, $2^\lambda < 2^{\lambda^+}$
or more strongly
if $\mu < \lambda$, $2^\mu < 2^\lambda$.)
That is are there arguments similar to the `iterative conception
of set' for why we should accept this principal?
One might think this was of just arithmetical interest. But axiom
E implies the Devlin Shelah weak diamond which implies Morley's
theorem for L_{omega_1, omega}. (If a sentence of this logic is
categorical up to all aleph_omega it is categorical in all
cardinalities.) Thus if one switches from first order to
infinitary logic a fundamental result of model theory requires a
slight extension of ZFC.
John T. Baldwin
Director, Office of Mathematics and Computer Education
Department of Mathematics, Statistics, and Computer Science
jbaldwin at uic.edu
312-413-2149
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607
Assistant to the director
Jan Nekola: 312-413-3750